nLab
fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos

Definition

For $(\Pi ⊣\Gamma ⊣\mathrm{LConst}):H\to \mathrm{\infty }\mathrm{Grpd}$ a locally ∞-connected (∞,1)-topos and $X\in H$ an object, we say that $\Pi (X)$ is the fundamental $\mathrm{\infty }$-groupoid of $X$ in $H$.

Properties

• This is the object that encodes the geometric homotopy groups in an (∞,1)-topos.

• For any $X\in H$ the $\mathrm{\infty }$-groupoid $\Pi (X)$ coincides with the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos (see there) of the over-(∞,1)-topos $H/X$.

Examples

• In the cohesive (∞,1)-topos Top the intrinsic fundamental $\mathrm{\infty }$-groupoid functor coincides with the ordinary fundamental ∞-groupoid of a topological space. See discrete ∞-groupoid for details.

• In ETop∞Grpd the intrinsic fundamental $\mathrm{\infty }$-groupoid is the generalization of that on Top to ∞-groupoids in paracompact spaces.

Related concepts

• geometric realization of cohesive ∞-groupoids